In my interpretation, the surface is somewhat like a sink that traps electric field line and therefore get charged in case there is charge inside.
A Gaussian surface is entirely passive and has no effect on the electric field and the charges which produce it.
Electric field lines are a good way of visualising an electric field and they "start" on positive charge and "end" on negative charges.
A (historical) way of thinking of electric flux is that it is a measure of the number of electric field lines passing through an area - the "flow" of the electric field through an area.
Consider a closed surface enclosing a volume.
If there are no charges inside the closed surface any electric field lines which enter the surface from outside the surface must also exit the surface as there is nowhere for the electric field lines to terminate, so the net electric flux though the closed surface is zero. (The number of electric field lines which enter through the surface is equal to the number of electric field lines which leave through the surface).
Positive charges alone inside the surface will produce electric field lines and so there will be an electric flux through the surface which might be counted as positive as the electric field lines are pointing out from the enclosed volume as they pass through the surface.
Negative charges alone inside the surface will produce electric field lines and so there will be a net electric flux though the surface but in this case the electric flux would be counted as negative as the electric field lines are pointing in towards the enclosed volume as they pass through the surface.
Why is electric flux through a closed surface with charge inside non zero?
Is not always true because within the enclosed volume there could be positive and negative charges of equal magnitude which would result in zero net electric flux through the enclosing surface.